Relationship between portfolio diversification and value at risk: Empirical evidence

This research explores the risk associated with the stocks prices in the seventeen selected companies that are listed in Indian BSE (100) National as well as portfolios of investment that are constructed from these seventeen companies employed. Additionally, for considering the possibility of international diversification, construction of portfolios of investment form stock price indexes in various emerging markets and developed countries of the world is considered. Correlations for domestically as well as internationally diversified portfolios are computed to unveil the relationship between stock prices of various firms as well as domestic and internationally diversified portfolios of investments. Further, to understand the effect of diversification on the risk associated with each of the portfolios of investments employed, value at risk analysis (VaR) is undertaken for studying the benefits associated with domestic as well as international diversification (if any).The study results show that domestic diversification lowers the expected losses associated with each of the domestic portfolios of investment employed where the international diversification substantially mitigates the portfolio risks. Results from VaR analysis reveal that diversification lowers the portfolio risks and additional reduction in portfolio risks is realized by international diversification.     

The dark side of Bitcoin: Do Emerging Asian Islamic markets help subdue the ethical risk?

Continuous financing of illicit activities (drug and human trafficking, child abuse, cybercrimes) through Bitcoin nurtures the ethical risk of investors. Building on this argument, the current study investigates the extreme tail dependence between Bitcoin and Emerging Asian Islamic (EAI) markets. We report multiple tail-dependent copulas differing across turmoil periods for the whole sample period. Under the ethical-risk hypothesis and modern portfolio theory, our findings demonstrated stronger safe-haven properties of EAIs for Bitcoin to mitigate ethical risk, and higher diversification benefits are documented for both equally adjusted and optimal portfolios. We formulated useful implications for policymakers, governments, regulation authorities, ethical investors, and portfolio managers for policymaking and strategizing their investment portfolios.     

A Markowitz Optimization of Commodity Futures Portfolios

We examine the diversification benefits of using individual futures contracts instead of simply a commodity index. We determine the ex‐ante, ex‐post, and stability results for optimal Markowitz portfolios, investigate the instability between the ex‐ante and ex‐post results, and compare our results to traditional and naïve portfolios. The ex‐ante complete futures portfolio dominates the traditional and naive portfolios and the ex‐post portfolio outperforms the naïve portfolio. The instability between the ex‐ante and ex‐post results is primarily driven by the time‐varying returns of the individual assets rather than by risk. Finally, the Sharpe portfolio results are essentially identical to the Markowitz results. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:343‐368, 2013     

Pitfalls of Downside Performance Measures with Arbitrary Targets

Downside performance measures relate above target returns with lower partial moments. They were developed to resolve restrictive assumptions of the classical Sharpe ratio. While the Sharpe ratio evaluates whether portfolios of a mutual fund and the risk‐free asset dominate passive portfolios of the benchmark and the risk‐free asset, this characteristic cannot be transferred to downside performance measures with arbitrary targets. We show that downside performance measures assign different values to passive benchmark strategies if the target differs from the risk‐free rate. This effect can lead to reverse rankings of financial assets. Therefore, downside performance measures are only applicable in asset management if the target is set equal to the risk‐free rate.     

COMMENT ON “SKEWNESS‐AWARE ASSET ALLOCATION”

This paper discusses risk measures proposed by Low et al. One of their new risk measures is skewness‐aware deviation, which is closely related to constant absolute risk aversion utility functions. This measure captures downside risk more effectively than traditional variance does. The authors also propose a second measure, skewness‐aware variance, which is derived from skewness‐aware deviation. This measure simplifies asset allocation problems and empirical results indicate that it captures risk better than traditional variance. However, this measure is also found to be inconsistent due to factor selection. Additionally, in the aspect of skewness‐aware deviation, optimal portfolios based upon skewness‐aware variance are sometimes less efficient than optimal portfolios that base themselves on traditional variance.     

Skewness,sampling risk,and the importance of diversification

Recent papers have extended portfolio theory to include skewness along with mean return and variance to explain security preferences. Because the positive skewness typically present in individual assets such as stocks and call options is rapidly reduced through diversification, several authors have suggested that a preference for positive skewness can lead to antidiversification as investors attempt to capture the greatest amount of positive skew. These analyses ignore the sampling risk (the probability that the skewness of a particular portfolio of size n will be near its expected value) that exists when less than fully diversified portfolios are held.This analysis presents skewness statistics for three security populations over three market periods to illustrate the relationship between sampling risk and diversification. Small portfolios of stocks or call options are shown to possess greater expected skewness than larger portfolios for each sample period examined, but small portfolios also exhibit the greatest sampling risk for skewness. Depending upon an investor's skewness preference, some diversification in these assets appears appropriate to increase the confidence regarding the actual skewness that will be observed for a particular portfolio chosen. Expected skewness for covered call portfolios is negative for all portfolio sizes and sample periods and declines as portfolio size inreases. The elimination of sampling risk and the reduction of negative expected skewness can be accomplished through complete diversification in these assets.     

TRACTABLE ROBUST EXPECTED UTILITY AND RISK MODELS FOR PORTFOLIO OPTIMIZATION

Expected utility models in portfolio optimization are based on the assumption of complete knowledge of the distribution of random returns. In this paper, we relax this assumption to the knowledge of only the mean, covariance, and support information. No additional restrictions on the type of distribution such as normality is made. The investor’s utility is modeled as a piecewise‐linear concave function. We derive exact and approximate optimal trading strategies for a robust (maximin) expected utility model, where the investor maximizes his worst‐case expected utility over a set of ambiguous distributions. The optimal portfolios are identified using a tractable conic programming approach. Extensions of the model to capture asymmetry using partitioned statistics information and box‐type uncertainty in the mean and covariance matrix are provided. Using the optimized certainty equivalent framework, we provide connections of our results with robust or ambiguous convex risk measures, in which the investor minimizes his worst‐case risk under distributional ambiguity. New closed‐form results for the worst‐case optimized certainty equivalent risk measures and optimal portfolios are provided for two‐ and three‐piece utility functions. For more complicated utility functions, computational experiments indicate that such robust approaches can provide good trading strategies in financial markets.     

MULTIVARIATE RISK MEASURES: A CONSTRUCTIVE APPROACH BASED ON SELECTIONS

Since risky positions in multivariate portfolios can be offset by various choices of capital requirements that depend on the exchange rules and related transaction costs, it is natural to assume that the risk measures of random vectors are set‐valued. Furthermore, it is reasonable to include the exchange rules in the argument of the risk measure and so consider risk measures of set‐valued portfolios. This situation includes the classical Kabanov's transaction costs model, where the set‐valued portfolio is given by the sum of a random vector and an exchange cone, but also a number of further cases of additional liquidity constraints. We suggest a definition of the risk measure based on calling a set‐valued portfolio acceptable if it possesses a selection with all individually acceptable marginals. The obtained selection risk measure is coherent (or convex), law invariant, and has values being upper convex closed sets. We describe the dual representation of the selection risk measure and suggest efficient ways of approximating it from below and from above. In the case of Kabanov's exchange cone model, it is shown how the selection risk measure relates to the set‐valued risk measures considered by Kulikov (2008, Theory Probab. Appl. 52, 614–635), Hamel and Heyde (2010, SIAM J. Financ. Math. 1, 66–95), and Hamel, Heyde, and Rudloff (2013, Math. Financ. Econ. 5, 1–28).     

Cross‐commodity analysis and applications to risk management

The understanding of joint asset return distributions is an important ingredient for managing risks of portfolios. Although this is a well‐discussed issue in fixed income and equity markets, it is a challenge for energy commodities. In this study we are concerned with describing the joint return distribution of energy‐related commodities futures, namely power, oil, gas, coal, and carbon. The objective of the study is threefold. First, we conduct a careful analysis of empirical returns and show how the class of multivariate generalized hyperbolic distributions performs in this context. Second, we present how risk measures can be computed for commodity portfolios based on generalized hyperbolic assumptions. And finally, we discuss the implications of our findings for risk management analyzing the exposure of power plants, which represent typical energy portfolios. Our main findings are that risk estimates based on a normal distribution in the context of energy commodities can be statistically improved using generalized hyperbolic distributions. Those distributions are flexible enough to incorporate many characteristics of commodity returns and yield more accurate risk estimates. Our analysis of the market suggests that carbon allowances can be a helpful tool for controlling the risk exposure of a typical energy portfolio representing a power plant. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:197–217, 2009     

The tail risk of emerging stock markets

We investigate tail risk in emerging stock markets at the country, regional and world levels, by comparing the investable and non-investable segments in terms of the expected shortfall of standardized returns and tail dependence on the world market. Employing the skewed Student-t GJR-GARCH model and the SJC copula, we show that most investable portfolios have lower tail risk but higher tail dependence than non-investable ones; emerging markets are likely more dependent on the world market during large joint losses than large joint gains; and tail dependence of the aggregate and investable markets on the world market varies across countries and regions.     

现代资产组合理论的一个拓展:出口市场组合模型——关于我国出口市场多元化的理论思考

出口多元化战略实施十多年以来,我国出口市场仍较为集中。出口多元化战略包括两层含义:产品的多元化和市场的多元化。国内理论界关于多元化战略的研究主要集中在前者,关于后者的研究则比较罕见。本文试图借鉴现代资产组合理论的思想,运用相对方差来衡量风险,构造了出口市场组合模型,用于定量分析多元化战略中的“市场多元化”问题,为降低我国产品出口所面临的市场风险提供理论指导。     

International bank lending: Political risk and portfolio diversification

The extent to which a large U.S. bank, all U.S. banks, and banks in the Group of Ten took account of political risk in their international country exposures in 1976 is tested using a simple portfolio diversification model. Assuming that political risks are important relative to economic risks, and that political risks are uncorrelated across countries, these banks' exposures should be negatively related to political risk indices. However, the portfolios of these banks appear to be related to political risk only insofar as political risk is roughly approximated by GNP per capita. International banks were not yet able to systematically vary their international portfolios with respect to political risk.     

Further evaluation of conglomerate performance

In the early years conglomerates were seen as the financial concept of the future. More recently their economic advantages have been seriously questioned. This article reviews the existing literature on conglomerate performance and extends the investigation through the early seventies. While conglomerates provide investors with less variability from market movements than do nonconglomerate firms, they provide less diversification than closed-end investment companies and mutual funds. Furthermore, risk adjusted performance measures of conglomerates did not differ significantly from those of other firms of portfolios. Thus, conglomerates should be viewed simply as another category of investments that plot along the security market line.     

Diversification patterns and survival as firms mature

We focus on the relationship between age and diversification patterns of German machine tool manufacturers in the post-war era. We distinguish between ‘minor diversification’ (adding a new product variation within a familiar submarket) and ‘major diversification’ (expanding the product portfolio into new submarkets). Our analysis reveals four main insights. First, we observe that firms have lower diversification rates as they grow older, and that eventually diversification rates even turn negative for old firms on average (where negative diversification corresponds to exit from certain product lines). Second, we find that product portfolios of larger firms tend to be more diversified. Third, with respect to consecutive diversification activities, quantile autoregression plots show that firms experiencing diversification in one period are unlikely to repeat this behavior in the following year. Fourth, survival estimations reveal that diversification activities reduce the risk of exit in general and to a varying degree at different ages. These results are interpreted using Penrosean growth theory.     

Efficient use of commodity futures in diversified portfolios

We provide evidence on the role of commodity futures in portfolios comprised of stocks, bonds, T‐bills, and real estate. Over the period investigated (1973–1997), Markowitz optimization over a range of risk levels gives substantial weight to commodity futures, thereby enhancing the portfolios’ returns. We find dramatically different results when we use a simple ex ante measure of monetary stringency to dichotomize the sample into expansive‐versus‐restrictive monetary‐policy periods. In periods characterized by restrictive monetary policy, commodity futures are shown to have substantial weight in the efficient portfolios, with significant return enhancement at all levels of risk. In periods characterized by expansive monetary policy, commodity futures are shown to have little or no weight in the efficient portfolios, with no return enhancement at all levels of risk. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:489–506, 2000     

RISK MEASURES: RATIONALITY AND DIVERSIFICATION

When there is uncertainty about interest rates (typically due to either illiquidity or defaultability of zero coupon bonds) the cash‐additivity assumption on risk measures becomes problematic. When this assumption is weakened, to cash‐subadditivity for example, the equivalence between convexity and the diversification principle no longer holds. In fact, this principle only implies (and it is implied by) quasiconvexity. For this reason, in this paper quasiconvex risk measures are studied. We provide a dual characterization of quasiconvex cash‐subadditive risk measures and we establish necessary and sufficient conditions for their law invariance. As a byproduct, we obtain an alternative characterization of the actuarial mean value premium principle.     

Geographic diversification and risk in banking

Tests of the hypothesis that geographic diversification affects bank risk are conducted on large samples of banking organizations (1976–1985) and focus on intrastate geographic diversification experience. Three composite measures of risk are included iin the tests along with the individual components of these measures. Results show that while composite measures of risk are reduced by geographic diverisification, some inidividual components of these measures increase. Importantly, the results show lower financial risk (the variation in earnings), which is predicted by portfolio theory. However, we also observe lower levels of earnings and capital with greater diversification implying, ceteris paribus, higher risk. This effect is not predicted by portfolio theory, but is predicted by our notion of operating risk. There is apparently more than pure financial risk involved with diversification by firms.     

COVID-19 and tail risk contagion across commodity futures markets

This paper examines the impact of COVID-19 on tail risk contagion across commodity futures markets using a copula-based network method. We document a significant increase in the lower and upper tail contagiousness of commodities following the COVID-19 outbreak. Contagion shows an obvious clustering characteristic, that is, there is higher tail risk connectedness between commodities in the same category. Agricultural commodities are significantly less contagious than metals and energy commodities; soft commodities in particular can offer investors significant diversification benefits. There are several hub commodities in the contagion network, chief among them copper, which are good transmitters of shocks and should be treated with caution by investors and regulators. Although tail risk and contagiousness of individual commodities increase together during the pandemic, we find a negative cross-sectional relationship between tail risk and contagiousness, that is, commodities with high tail risk are not necessarily highly contagious and may even be less so.     

Emerging markets,downside risk and the asset allocation decision

This study examines the use of downside risk measures in the construction of an optimal international portfolio, with particular reference to the estimated allocations in emerging markets and the out-of-sample performance of the optimal portfolios. The use of downside risk measures is assessed due to the problems of using a conventional mean-variance analysis approach in the presence of the non-normality often found to be present in emerging market data. The data set used consists of the MSCI indices for developed equity markets and the IFC data set on emerging markets. The primary component of the paper consists of the construction of optimal portfolios under both mean-variance and downside risk frameworks. In addition, the use of Bayes–Stein estimators is also assessed, in an attempt to reduce estimation error. The resulting estimated allocations are then used to assess the out-of-sample performance of the optimal portfolios. The results indicate that for risk-averse investors the use of downside risk measures can result in significant improvements in performance.     

LARGE DEVIATIONS IN MULTIFACTOR PORTFOLIO CREDIT RISK

The measurement of portfolio credit risk focuses on rare but significant large-loss events. This paper investigates rare event asymptotics for the loss distribution in the widely used Gaussian copula model of portfolio credit risk. We establish logarithmic limits for the tail of the loss distribution in two limiting regimes. The first limit examines the tail of the loss distribution at increasingly high loss thresholds; the second limiting regime is based on letting the individual loss probabilities decrease toward zero. Both limits are also based on letting the size of the portfolio increase. Our analysis reveals a qualitative distinction between the two cases: in the rare-default regime, the tail of the loss distribution decreases exponentially, but in the large-threshold regime the decay is consistent with a power law. This indicates that the dependence between defaults imposed by the Gaussian copula is qualitatively different for portfolios of high-quality and lower-quality credits.